Control Theory Question

Hi there,

While reading an introductory text on control, the author stated that for all physical systems, the number of zeores is less than or equal to the number of poles. I want to know the reason behind that. Could anyone of the learned fellows help me with the answer?

Thanks in advance.

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A naked zero in a system is tantamount to having a naked differentiator. These things simply haven't been observed in nature. There's no theory to support this; like the first law of thermodynamics and the observation of the conservation of mass it's based on the observation that they've never been seen and that the universe would be a very different place if they could exist, so we conclude that they aren't there.

In physical systems the underlying "things" that drives the system dynamics always act to integrate some other quantity, and each individual integrator creates a "state" in the system. If you do have a system output that acts like a differentiator it's because there's a state that it's comparing against (either by the purposeful intent of man or just because that's how the system behaves), and the differentiation ends up being band-limited.

If you could have zeros without poles in a physical system then you could either get infinite power out with finite power input in (by extending the frequency response to infinity), or you could predict the future, or both.

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Tim Wescott

Hi Tim,

Thank you very much for your comments. In fact, after posting the question, I took a hypothetical system with 3 zeros and 2 poles. After a little bit of methamatics, I came to have a , as you said, naked differentiator in the output expression. I could reason that this is not something common, as it would drive many systems unstable, if lets say, a signal is injected in the input whose derivative is infinity, or simply undefined.

Thanks a lot.

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